Hello, I am looking into proving that the Chinese Remainder Theorem will work for two pairs of congruences IFF a congruent to b modulo(gcd(n,m)) for x congruent to a mod(n) and x congruent to b mod(m). I have gotten one direction, that given a solution to the congruences mod(m*n), then a congruent to b mod(gcd(m,n)). My issue is going the other way, given a congruent to b mod(gcd(m,n)), show a solution to the congruences exists mod(m*n). Can anybody help me with a start? I tried expressing the relationship between a and b and using that to determine what x would have to be, but I'm not convinced of the results.